On an estimate of Axler and Shapiro (Q1061276)

Let D be a bounded strictly pseudoconvex domain in \({\mathbb{C}}^ n\) with smooth boundary. The main result is an estimate of the form \[ dist(f,VMOA)\leq 2 dist(\bar f,H^{\infty}+C),\quad f\in H^{\infty}, \] where the second distance is measured in the supremum norm, and the first distance is measured in the norm of any one of various BMO-type spaces associated with D (here \(H^{\infty}+C\) is the linear span of \(H^{\infty}\) and the space C of continuous functions on the boundary). As a consequence, a related estimate of \textit{S. Axler} and \textit{J. H. Shapiro} [Math. Ann. 271, 161-183 (1985; Zbl 0541.30021)] is seen to extend from the unit ball to strictly pseudoconvex domains.